Theorem sectss

Description: The section relation is a relation between morphisms from X to Y and morphisms from Y to X . (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	issect.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	issect.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	issect.o	⊢ · = ( comp ‘ 𝐶 )
	issect.i	⊢ 1 = ( Id ‘ 𝐶 )
	issect.s	⊢ 𝑆 = ( Sect ‘ 𝐶 )
	issect.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	issect.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	issect.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	sectss	⊢ ( 𝜑 → ( 𝑋 𝑆 𝑌 ) ⊆ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		issect.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		issect.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
3		issect.o	⊢ · = ( comp ‘ 𝐶 )
4		issect.i	⊢ 1 = ( Id ‘ 𝐶 )
5		issect.s	⊢ 𝑆 = ( Sect ‘ 𝐶 )
6		issect.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
7		issect.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8		issect.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
9	1 2 3 4 5 6 7 8	sectfval	⊢ ( 𝜑 → ( 𝑋 𝑆 𝑌 ) = { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } )
10		opabssxp	⊢ { ⟨ 𝑓 , 𝑔 ⟩ ∣ ( ( 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ∧ 𝑔 ∈ ( 𝑌 𝐻 𝑋 ) ) ∧ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑋 ) 𝑓 ) = ( 1 ‘ 𝑋 ) ) } ⊆ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) )
11	9 10	eqsstrdi	⊢ ( 𝜑 → ( 𝑋 𝑆 𝑌 ) ⊆ ( ( 𝑋 𝐻 𝑌 ) × ( 𝑌 𝐻 𝑋 ) ) )